Variational approximation of a functional of Mumford-Shah type in codimension higher than one

submittedyear: 2013 journal: ESIAM: COCV abstract: In this paper we consider a new kind of Mumford-Shah functional $E(u,\Omega)$ for maps $u:\mathbb{R}^m\rightarrow \mathbb{R}^n$ with $m\geq n$. The most important novelty is that the energy features a singular set $S_u$ of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy $E(u,\Omega)$ via $\Gamma-$convergence, in the same spirit of the work by Ambrosio and Tortorelli.